Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator
It is shown that the application of the model of the hydrogen atom, which is based on the theory of the elliptic oscillator, makes it possible to describe the structure of electron orbits. The investigation is based on asymptotics methods of nonlinear mechanics. It was established that the stable...
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2001
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irk-123456789-1746952021-01-28T01:26:30Z Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator Santagata, C. It is shown that the application of the model of the hydrogen atom, which is based on the theory of the elliptic oscillator, makes it possible to describe the structure of electron orbits. The investigation is based on asymptotics methods of nonlinear mechanics. It was established that the stable orbits of the electron correspond to certain resonant states. A numerical simulation of the problem was carried out 2001 Article Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator / C. Santagata, C. Santagata // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 376-388. — Бібліогр.: 7 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174695 en Нелінійні коливання Інститут математики НАН України |
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It is shown that the application of the model of the hydrogen atom, which is based on the theory
of the elliptic oscillator, makes it possible to describe the structure of electron orbits. The investigation is based on asymptotics methods of nonlinear mechanics. It was established that the
stable orbits of the electron correspond to certain resonant states. A numerical simulation of
the problem was carried out |
| format |
Article |
| author |
Santagata, C. |
| spellingShingle |
Santagata, C. Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator Нелінійні коливання |
| author_facet |
Santagata, C. |
| author_sort |
Santagata, C. |
| title |
Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator |
| title_short |
Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator |
| title_full |
Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator |
| title_fullStr |
Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator |
| title_full_unstemmed |
Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator |
| title_sort |
nonlinear simulation of the hydrogen atom based on the model of the elliptic oscillator |
| publisher |
Інститут математики НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/174695 |
| citation_txt |
Nonlinear Simulation of the Hydrogen Atom Based on the Model of the Elliptic Oscillator / C. Santagata, C. Santagata // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 376-388. — Бібліогр.: 7 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT santagatac nonlinearsimulationofthehydrogenatombasedonthemodeloftheellipticoscillator |
| first_indexed |
2025-07-15T11:44:46Z |
| last_indexed |
2025-07-15T11:44:46Z |
| _version_ |
1837713207396925440 |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 3, 2001
NONLINEAR SIMULATION OF THE HYDROGEN ATOM BASED ON THE
MODEL OF THE ELLIPTIC OSCILLATOR
C. Santagata
Second University of Naples, Aversa, Italy
It is shown that the application of the model of the hydrogen atom, which is based on the theory
of the elliptic oscillator, makes it possible to describe the structure of electron orbits. The investi-
gation is based on asymptotics methods of nonlinear mechanics. It was established that the
stable orbits of the electron correspond to certain resonant states. A numerical simulation of
the problem was carried out.
AMS Subject Classification: 34E10
1. Introduction
The theories by Bohr, Sommerfeld, Heisenberg and Schrödinger [1 – 5] follow directly or
indirectly from the Planck’s postulate E = hνn. It is possible to show that, still using this
postulate as a starting point, one can obtain other alternate quantum theories, whose reliability
is the same as the previous ones. If we do not use the Planck’s postulate and consider the nonli-
nearity of the electric field, that solves even the black body problem from the point of view of
the classical continuity, it is possible to show that the equation ẍ+γẋ+ω2x+ξx2 = δ cos(Ωt+ϕ),
where γ is the classical radiation pressure and ϕ = 2/(3 · 137), describes correctly the hydrogen
stationary states, and further forecasts new lines forbidden by the previous approaches. Finally,
the stationary states of matter are interpreted in the clear classical environment as discrete
resonance states between the electron bounded to the atom and even fully interacting with the
surrounding environment.
2. The Nonlinearity of Coulomb’s Law and Its Implications
Usually investigations on electron orbits were based on the model of a circular orbit, which
results in a model of linear two-dimensional oscillator or the Heisenberg model, which includes
postulation of the nonlinear term
d2x
dt2
+ ω2x+ λx2 = 0.
Based on these models it is possible to construct energy levels by means of the Plank’s
hypothesis (E = hνn).
In order to study the hydrogen atom without making recourse to Planck’s postulate, it is
necessary to highlight a fundamental feature of the electric field, i.e. its strong nonlinearity,
and all its important consequences. It is easy to show that in most deductive reasonings of
theoretical physics the nonlinearity of Coulomb’s law has been always neglected.
Let us consider now the ellipse represented by the equation (ε < 0)
376 c© C. Santagata, 2001
NONLINEAR SIMULATION OF THE HYDROGEN ATOM BASED .. . 377
r =
p
1 + ε cosα
. (1)
From (1) by setting cosα = x/r, we deduce that r = p(1 − εx/p). This implies that the
projection of the force along the abscissae axis is given by
Fx = −e
2
r3
x = − e2
p3(1− εxp )3
x.
Therefore, denoting
ω2 =
e2
mp3
,
we finally obtain the equation along the x and y axes,
d2x
dt2
+
ω2x(
1− εxp
)3 = 0,
d2y
dt2
+
ω2y(
1− εxp
)3 = 0.
Contrary to the classical approach, this procedure makes it possible to find the force-defor-
mation law of the elliptic oscillator equation.
All these considerations imply that, to replicate the Rutherford’s model with major accuracy,
under the hypothesis that the external perturbation would not modify abruptly the original
orbit, it should be necessary to consider, for example in the Planck’s case, the equation
d2x
dt2
+ γ
dx
dt
+
ω2x(
1− εxp
)3 =
1
e
√
2π
+∞∫
−∞
E(Ω) exp(iΩt)dΩ, (2)
that, up to small nonlinearity, coincides with the equation of Heisenberg. Indeed we get
ω2x
(
1− εx
p
)−3
' ω2x
(
1 + 3ε
x
p
)
= ω2x+ 3
ε
p
ω2x2 = ω2x+ λx2.
The fact that, in the classical study of wave-particle interaction phenomena, the atom has always
been represented by a linear model has clear and strong negative consequences. Actually, the
transfer of energy from the wave to the matter, given the model adopted by Planck, occurs
exclusively by means of the fundamental resonance phenomenon. When the nonlinearity is
taken into account, it is well-known [6, 7] and easy to verify, that the multiresonance phenomenon
occurs. Actually the resonance phenomenon occurs only when the external pulsation Ω tends to
the natural pulsation ω of the system considered, but occurs even when Ω/ω = n or ω/Ω = n
(this single fact would justify physically and would extend Planck’s postulate). Therefore, when
378 C. SANTAGATA
the nonlinearity of the electric field is neglected, we have a very limited forecast about the reali-
ty, and this is the main problem of a number of failures generally attributed to inner defects
of classical mechanics (namely of its part relative to formulas deduced by the study of si-
ngle central fields only). It is enough to consider that the harmonic oscillator can be obtained
from the anharmonic oscillator by taking into account only the first term of an infinite series
development. In this last case, the analytic forecast of the existence of infinite resonances can
be justified intuitively by noting that when the distance from the central charge increases we
have a series of linear oscillators with varying values of ω2 = e2/(md3), so that a specific and
distinct resonance frequency can be associated to each of these values.
In the following we will study the equation
d2x
dt2
+ γ
dx
dt
+ ω2x+ 3
ε
p
ω2x2 = δ exp(−iΩt), (3)
but will also consider the more complete equation,
d2x
dt2
+ γ
dx
dt
+
ω2x(
1− εxp
)3 = δ exp(−iΩt). (4)
We will see that the results of determining the electron orbits obtained by each of these two
equations are completely analogous. Actually, as we can see by observing equation (4), we
must study the possible stability and instability conditions of an electron that, in the absence
of radiation emission and of interaction with the surrounding environment (represented by the
external force), would describe, according to classical mechanics, an elliptic orbit.
3. Existence of a Linear Stationary Solution
When an electromagnetic wave hits the above-mentioned nonlinear oscillator, we can find
a variety of solutions. It is known that they depend exclusively on initial conditions, which can
give rise to such a great number of solutions that they could seem to be chaotic. As an example
of the possible solutions that generally could occur, we can cite the case in which the peripheral
particle, instead of orbiting around the central charge, oscillates harmonically along a small
segment orthogonal to one of the equipotential surfaces generated by the central charge, with
an energy loss due to the radiation emitted exactly equal to the energy absorbed from the
external environment (stationary solution). This case would not be physically understandable
if we still thought of a completely isolated atom.
With the hypothesis that |ε| < 1 we can write relation (4) in the following form:
d2x
dt2
+ γ
dx
dt
+ ω2x+ 3ω2 ε
p
x2 + 6ω2 ε
2
p2
x3 + 10ω2 ε
3
p3
x4 + . . . = δ exp(−iΩt). (5)
Now let us assume that there exists a solution of the type
x(t) =
∞∑
n=1
An exp(−inΩt). (6)
NONLINEAR SIMULATION OF THE HYDROGEN ATOM BASED .. . 379
Substituting (6) into (5) and assuming thatAi >> Ai+1, we find the following values for various
coefficients of the series (6):
A1 =
δ
Ω2 − ω2 + iγΩ
, A2 = 3
ω2εA2
1
p(4Ω2 − ω2 + 2iγΩ)
, A3 = 6
ω2εA1(A2p+ εA2
1)
p2(9Ω2 − ω2 + 3iγΩ)
,
(7)
A4 =
εω2(6A1A3p
2 + 3A2
2p
2 + 18εA2
1A2p+ 10ε2A4
1)
p3(16Ω2 − ω2 + 4iγΩ)
, . . . , An =
εω2(F (Ai, p, ε))
(n2Ω2 − ω2 + niγΩ)
.
We can observe that the expression of the generic term An (excluding the ε variable and the
resonant term 1/(n2Ω2 − ω2 + niγΩ)) is practically coincident with the corresponding terms
which are obtained for the Heisenberg’s equation for relevant xi.
When we substitute A1 into the subsequent terms we get
Ψ1 = − δ
Ω2 − ω2 + iγΩ
exp(−iΩt),
Ψ2 = 3
ω2δ2ε
p(Ω2 − ω2 + iγΩ)2(4Ω2 − ω2 + 2iγΩ)
exp(−2iΩt),
(8)
Ψ3 = −12
ω2δ3ε2(ω2 + 2Ω2 + iγΩ)
p2(Ω2 − ω2 + iγΩ)3(4Ω2 − ω2 + 2iγΩ)(9Ω2 − ω2 + 3iγΩ)
exp(−3iΩt),
Ψn = . . .
Even in this case we observe that the generic term in the previous series is resonant both
at its own pulsation and at all previous critical pulsations. Furthermore we can see that this
solution, expressed in terms of harmonics of the type (nΩ), shows a resonance phenomenon
only if the condition
ω = Ωn (9)
is satisfied. Going into the field of real numbers, we have
Ψ1 = −
δ
(
Ω2 − ω2
)
(Ω2 − ω2)2 + γ2Ω2
cos(Ωt) +
δγΩ
(Ω2 − ω2)2 + γ2Ω2
sin(Ωt),
380 C. SANTAGATA
Ψ2 = 3
ω2εδ2
((
ω2 − Ω2
)2 − γ2Ω2
) (
4Ω2 − ω2
)
− 2ω2εδ2
(
2Ω3γ − 2ω2γΩ
)2
γΩ((
(ω2 − Ω2)2 − γ2Ω2
)2
+ (2Ω3γ − 2ω2γΩ)2
)
p
(
(4Ω2 − ω2)2 + 4γ2Ω2
)
cos(2Ωt)
+3
−ω2εδ2
(
2Ω3γ − 2ω2γΩ
)2 (
4Ω2 − ω2
)
− 2ω2εδ2
((
ω2 − Ω2
)2 − γ2Ω2
)
γΩ((
(ω2 − Ω2)2 − γ2Ω2
)2
+ (2Ω3γ − 2ω2γΩ)2
)
p
(
(4Ω2 − ω2)2 + 4γ2Ω2
)
sin(2Ωt),
(10)
. . . . . . . . .
Ψn =
An
(n2Ω2 − ω2)2 + n2γ2Ω2
cos(nΩt) +
Bn
(n2Ω2 − ω2)2 + n2γ2Ω2
sin(nΩt).
For obvious problems of space, in the last equation we have shown only the resonance nΩ,
but it is important to remember that each term of the series also contains resonances (n− 1)Ω,
(n − 2)Ω, ... . Therefore we can also have amplitudes Ψn,n−1, Ψn,n−2, . . . ,Ψn,1, besides Ψn,n.
This concept can be expressed synthetically by a triangular matrix that embodies the features
of this oscillator, ∣∣∣∣∣∣∣∣∣∣
Ψ1,1
Ψ2,1 Ψ2,2
Ψ3,1 Ψ3,2 Ψ3,3
. . .
Ψn,1 Ψn,2 Ψn,3 Ψn,n
∣∣∣∣∣∣∣∣∣∣
. (11)
Actually, in the first row of this matrix, which expresses features of the first term of the series
development, there is only the amplitude Ψ1,1 due to the presence of the main resonance only;
in the second row, which corresponds to the second term of the development, there are the first
and second resonance, in the third row there are the first, second, and third resonances, and
so on.
In the following, as usual, we will use the expression elastic amplitude to denote the coeffici-
ent of cos(nΩt) and the expression absorption amplitude to denote the coefficient of sin(nΩt),
since this is the term that generates a speed in phase with the driving force.
Thus, a particular solution of equation (5) can be written in the form
x(t) =
∞∑
n=1
Ψi. (12)
The terms in series (12) are connected with the corresponding equations of system (10) has the
same aspect of the stationary solution of the damped harmonic oscillator which is of the type
Ψ1 = B1 cos Ωt+ C1 sin Ωt, Ψ2 = (B2 cos 2Ωt+ C2 sin 2Ωt)ε,
. . . , Ψn = (Bn cosnΩt+ Cn sinnΩt)εn. (13)
NONLINEAR SIMULATION OF THE HYDROGEN ATOM BASED .. . 381
This implies that the solution of our equation can also be written in the following form
x =
∞∑
n=1
ρn cos(nΩt− Φn), (14)
where the following definitions have been used (note that each oscillator has a distinct phase
that characterizes it further),
ρn =
√
B2
n + C2
n, tan Φn =
Cn
Bn
.
The considerations above lead us to the conclusion that, when secular terms are absent,
relations (6) or their equivalent relation (14) represent a stationary solution of equation (5). It
is reasonable to assume that it is also unique, due to the univocal determination of the terms
Ai. Furthermore, we can observe that the solution we have found can be interpreted as a sum
of the stationary solutions of an infinite set of damped harmonic oscillators driven by proper
external forces and correlated with each other.
4. Evaluation of the Series Coefficients
Relations (7), in the hypothesis that ε is sufficiently smaller than 1, can be simplified by
neglecting the terms depending on ε2, and in such a case, they become
A1 =
δ
Ω2 − ω2 + iγΩ
, A2 = 3
ω2εA2
1
p(4Ω2 − ω2 + 2iγΩ)
, A3 = 3
ω2ε2A1A2
p(9Ω2 − ω2 + 3iγΩ)
,
(15)
A4 = 3
ω2ε(2A1A3 +A2
2)
p(16Ω2 − ω2 + 4iγΩ)
, A5 = 3
ω2ε(2A1A4 + 2A2A3)
p(25Ω2 − ω2 + 5iγΩ)
, . . . .
We want to establish an algorithm that enables us to calculate any term of series (15). Let
us consider, for example, the fourth term. If we define the two matrices
A = |A1 A2 A3| and B =
∣∣∣∣∣∣
A3
A2
A1
∣∣∣∣∣∣ ,
where B is the transposed and reversed matrix of A, we have
|A| × |B| = 2A1A3 +A2
2.
We can also verify that this rule, if we accept a less precise solution, is valid for any coeffici-
ent of the series. Therefore, we can write in short
x(t) =
∞∑
n=1
3ω2ε
p (n2Ω2 − ω2 + niγΩ)
|A1 A2 A3...An−1| ×
∣∣∣∣∣∣∣∣∣∣
An−1
. . .
A3
A2
A1
∣∣∣∣∣∣∣∣∣∣
exp(−niΩt), (16)
382 C. SANTAGATA
where the initial term is
A1 =
δ
Ω2 − ω2 + iγΩ
,
and the symbol × has been used for the matrix product (that in this case does not commute).
5. Graphical Representations
We will limit ourselves to considerations concerning the first four terms of series (10), and
will set the following values1:
(ε = 0.8; p = 30; δ = 300; ω = 16; γ = 0.1) .
With these choices we find the following 3D-graph for t ∈ [0, 3] and Ω ∈ [3.5, 10]. Since the
critical pulsation of the oscillator is equal to 16, and since we have considered only the first four
terms, we will have a resonance condition when
(Ω = 16; Ω = 8; Ω = 5.333; Ω = 4) .
Fig. 1. Miscellaneus resonances of an elliptic oscillator.
In Fig. 1 above, we can see the peaks associated to the three resonances Ω = 4, Ω = 5.33,
and Ω = 8. The peak at Ω = 16 is not represented, since it is outside the range chosen for Ω.
Thus, if the present model is valid, there will be an energy transfer from the driving force to
the oscillator only when the above mentioned resonances occur. In any other case, the external
wave goes through the system leaving it practically unperturbed.
Moreover, it is important to observe the following. Each component of the series contains
its proper resonance pulsation nΩ, and also contains all the previous (n− 1) critical pulsations.
Thus, the resonance nΩ also appears in all the subsequent terms, and, therefore, when the
driving force stimulates this resonance, it activates all the infinite oscillators n, n+ 1, n+ 2, . . . .
Thus the energy, represented by the square of the amplitude, will distribute with a certain law
1These values are taken conventionally for qualitative demonstration of the approach potential and do not
correspond to the real physical values.
NONLINEAR SIMULATION OF THE HYDROGEN ATOM BASED .. . 383
among these oscillators. Noting that Ai is already much greater than Ai+1, if the nonlinearity
parameter ε is small, the following inequality
(Anε
n)2 >>
(
An+1ε
n+1
)2
will be certainly satisfied. This can be illustrated by Fig. 1 that was obtained for a moderately
high eccentricity. The figure also shows that the second resonance that occurs for Ω = 8 generates
amplitudes much higher than those relative to the third resonance that arises for Ω = 5.3. This
resonance, in turn, is much higher than the fourth resonance, which corresponds to Ω = 4.
a b c
Fig. 2. Envelopment amplitude.
This implies that, if the driving force stimulates the n-th term of the series and if ε is very small,
it is possible to set, with good approximation,
(x(t))2 ' (Ψn)2 . (17)
On the contrary, if the nonlinearity is higher (ε ' 1), the various resonance peaks sums
and produce a composite profile, due to the subsequent terms that are no more negligible. This
effect can be understood by observing the graph in Fig. 2a, that is relative to the fundamental
resonance Ω = 16 that appears in all infinite terms of the series.
In Fig. 2b, which corresponds to the high pressure, we can notice various crests due to infini-
tely many subsequent terms of the series that resonate when Ω = ω. These crests have a quite
large mean width due to the fact that the value γ of the pressure is quite high.
On the contrary, if we lower the pressure γ to small values, the above mentioned crests
transform into very subtle and very high lines, as can be seen in Fig. 2c.
The fact that a single vibrating particle can assume the aspect of a boundless line spectrum
is singular.
6. Evaluation of Energy Levels
Of course, since now we don’t start from Planck’s hypothesis, we will find completely general
theoretical relations that, for the description of the hydrogen atom, will require that some
constants of the equation assume special values. On the other hand, it is easy to see that, wi-
thout prejudice for the precision of our calculations, we can consider the much more simple
equation,
ẍ+ γẋ+ ω2x+ 3ω2 ε
p
x2 = δ cos (Ωt+ ϕ) , (18)
384 C. SANTAGATA
which is analogous to the Heisenberg equation2 and is suitable to be solved with simple calculati-
ons whose extension is in any case immediate and in agreement with the results already obtai-
ned.
In the following we will limit ourselves only to the determination of the stationary levels
of Bohr and to an analytical outline of the emission process. To simplify the calculations as
much as possible (calculations that can easily be extended to formulas containing more than
one nonlinear term), we shall assume that there exists a solution of the type
x = A1 cos(Ωt) +A2 cos(2Ωt). (19)
After substituting this form of the solution in equation (18) and by equating values of the
same order of smallness we obtain the following solution:
A1 =
δ√
(ω2 − Ω2)2 + γ2Ω2
, A2 = −3
2
ω2εA2
1
p (ω2 − 4Ω2)
. (20)
We want to evaluate the work done by the external force. Multiplying (18) by (dx) and integra-
ting over the period (2π/Ω), (18) and (19) will give us∫
vdv + ω2
∫
xdx+
∫
3ω2 ε
p
x2dx+ γΩ2
∫
(A1 sin(Ωt) + 2A2 sin(2Ωt) + . . .)2 dt =
= −Ωδ
∫
cos(Ωt+ ϕ) (A1 sin(Ωt) + 2A2 sin(2Ωt) + . . .) dt. (21)
The first three integrals of (21) are equal to zero owing to the periodicity, so we have
γΩ2
∫ (
A1 sin(Ωt) + 2A2 sin(2Ωt) + . . .
)2
dt
= −Ωδ
∫
cos(Ωt+ ϕ) (A1 sin(Ωt) + 2A2 sin(2Ωt) + . . .) dt,
from which it follows that
γπ
Ω
(
A2
1 (1Ω)2 +A2
2 (2Ω)2 +A2
3 (3Ω)2 + . . .
)
= πδA1 sinϕ.
Therefore,
1
2
m
[
A2
1 (1Ω)2 +A2
2 (2Ω)2 +A2
3 (3Ω)2 + . . .
]
=
1
2
m
δ2
γ2
sin2 ϕ. (22)
This equation sets relation between the amplitudes Ai and their corresponding frequencies.
2Even Heisenberg used, at the beginning, the damping term, but subsequently, to simplify the calculations, he
eliminated it. In his treatments the external force is always absent because probably he realized that the use of
Planck’s postulate eliminated the orbit stability problem.
NONLINEAR SIMULATION OF THE HYDROGEN ATOM BASED .. . 385
Let us consider another interesting property of nolinearity manifestation. If we assume that
there is a driving force of the type
F =
n∑
j=1
δj exp(−inΩjt),
we can say that the terms activated will be only the corresponding terms relative to the resonances
Ωj and their linear combinations. In fact the equation
ẍ+ ω2x+ λx2 = δ1 cos (Ω1t) + δ2 cos (Ω2t)
admits the stationary solution
x = − λ
2ω2
(A2
1 +A2
2) +A1 cos (Ω1t) +A2 cos (Ω2t) +
λA2
1 cos (2Ω1t)
2(4Ω2
1 − ω2)
+
λA2
2 cos (2Ω2t)
2(4Ω2
2 − ω2)
+
λA1A2 cos (Ω1 − Ω2) t
((Ω1 − Ω2)
2 − ω2)
+
λA1A2 cos (Ω1 + Ω2) t
((Ω1 + Ω2)
2 − ω2)
with
A1 =
δ1
ω2 − Ω2
1
, A2 =
δ2
ω2 − Ω2
2
.
It is easy to see that, besides the occurrence of resonances at ω = Ω1, ω = Ω2, ω = 2Ω1, and
ω = 2Ω2, we will also have resonances at ω = Ω1+Ω2 and ω = Ω1−Ω2. This phenomenon has a
resemblance with the addition and subtraction principle discovered by W. Ritz in the hydrogen
atom spectrum.
7. Stability and Instability
Let us consider, for the sake of simplicity, the following equation3
d2x
dt2
+ γ
dx
dt
+ ω2x+ 3ω2 ε
p
x2 = δ cos(Ωt+ ϕ),
and let us assume that there exists a solution of the type
x(t) = A1 cos(Ωt) +A2 cos(2Ωt).
Substituting this solution in the first equation, collecting the terms containing sines and cosines,
and requiring that they must be equal to zero, we find the following identities:
1
2
−2A1Ω
2p+ 6ω2εA1A2 + 2ω2A1p
p
= δ cosϕ,
3It is necessary to note that this equation contains only terms of the second order in comparison with equati-
on (5). Neglecting the terms of the third order can essentially influence the inclination of the amplitude-frequency
characteristics. On the other hand we focused our attention on studying the equation similar to the Heisenberg one
for showing its basic regularities.
386 C. SANTAGATA
a b c
Fig. 3. Amplitude-frequency characteristics.
A2 =
3
2
ω2εA2
1
p(4Ω2 − ω2)
, γA1Ω = δ sin(ϕ).
Substituting the second equation into the first, squaring and summing the remaining equations,
we find [
(ω2 − Ω2) +
9
2
ω4ε2A2
1
p2 (4Ω2 − ω2)
]2
+ (γΩ)2 =
(
δ
A1
)2
. (23)
We have studied this implicit function for the following values:
ω = 16; ε = 0.8; p = 30; γ = 8; δ = 100,
and have obtained a graph shown in Fig. 3a.
Let us assume that we are able to increase gradually the pulsation Ω starting from the value
14. The corresponding ordinate is forced to trace the ascending branch of the curve. Let us
consider a small sphere that moves along this branch because of a hypothetical gravitational
force. We can imagine that, after having moved it along the mentioned branch, the small sphere
is let free just when it is very near to the abscissa Ω = 18. Then the sphere, being forced to
describe the curve because of the above mentioned gravitational force, will fall again towards
the potential well that corresponds to the resonance Ω = 8, as can be seen clearly in Fig. 3b.
On the contrary, if our small sphere is let free just at the highest point of the curve A, then
it can both return towards the resonance Ω = 8 or fall vertically onto the lower branch of the
curve at the right side of the previously mentioned highest point. This vertical fall A-B is due
to the fact that our external action forces a pulsation increase while the sphere, in order to
describe the broken part of the curve, should cause on the contrary a decrease of the pulsation.
Conversely, starting from a pulsation equal to 20 and imposing a gradual decrease of its
value, the particle will describe the above-mentioned curve until it reaches the pulsation Ω =
16.4 shown in Fig. 3b, and subsequently, for the same reason as described above, will be forced
to jump up to the higher branch.
It is evident that the particle, when the external variation of the pulsation disappears abruptly,
will remain in a vicinity of another resonance condition, giving back, under the form of radiati-
on, the energy gap between the two stationary states. Returning back to Fig. 3a, we can say that,
NONLINEAR SIMULATION OF THE HYDROGEN ATOM BASED .. . 387
starting from the pulsation Ω = 14, the small sphere describes the upper curve following the
arrows, until it reaches A. In this point it should return back, but the pulsation increase forbids
this, so it falls vertically into B, and then continues its movement on the right side of the lower
curve. On the other hand, going on with a continuous decrease of the pulsation, starting from
Ω = 20, the small sphere follows the path marked by the arrows below the curve, and thus,
once reached the point C, it leaves the curve and moves vertically until it reaches the upper
curve in the point D, continuing then towards the left side. These forecasts, besides describing
the shift of the lines due to the radiation pressure, also show that in the vicinity of the resonance
frequencies there are abrupt amplitude variations due to said instability phenomena.
Our curves are extremely sensitive to pressure variations. A pressure increase transforms
the graph shown in Fig. 3b into the graph shown in Fig. 3c.
In this figure we see the line corresponding to the resonance Ω = 8 that should reach an
infinite value. Of course this value is due to the fact that we have studied a solution by truncating
the series in an excessive way.
What we have said above, besides giving a justification to the shifts of spectrum lines, even
implies that the position occupied by the particle in a point of the space should be given by a
function indissolubly related to the value that the radiation pressure assumes in that point. Of
course this is a microscopic description of the reality that can be adapted to the hydrogen atom
with a procedure totally similar to the procedure used above. Therefore, in principle, the fact
that the atom has just a specific size should be related to the value that the radiation pressure
has at that point.
It could seem that there is a conflict between the former solution represented by relation
(20), that we will denote explicit solution, and the present implicit (and more precise) solution.
However, it is easy to see that, while (20) is an explicit solution of the differential equation
that makes it possible to determine various coefficients of the series, the solution adopted in
the present case does not have this property. The conflict disappears when we observe that (23)
makes it possible to write explicitly the term A1 only if we set ε2 = 0. In such a case we get
A1 =
δ√
(ω2 − Ω2) + γ2Ω2
,
and this expression is the same as that given by the explicit solution (20).
8. Acknowledgements
The author wants to express special gratitude and deep thankfulness to the physicist Prof.
Franco Selleri of the University of Bari, Italy, who, with his exemplary and particular criticism
to the article Vibration Mechanics: Atoms, Nuclei and Resonances, to be published soon, has
originated the present improvement of that paper.
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388 C. SANTAGATA
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Received 12.02.2001
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